The acceleration due to gravity decreases by Δg1 when a body is taken to a small height h << R. The acceleration due to gravity Δg2 decreases when the body is taken to a depth h fr... The acceleration due to gravity decreases by Δg1 when a body is taken to a small height h << R. The acceleration due to gravity Δg2 decreases when the body is taken to a depth h from the surface of the earth. Then (R = Radius of the earth)...
Understand the Problem
The question is asking about the effects of height and depth on the acceleration due to gravity. Specifically, it compares how the acceleration decreases when a body is taken to a small height above the Earth's surface versus when it is taken to a depth below the surface. The goal is to derive a relationship between the changes in gravity in these two scenarios.
Answer
The relationship is given by $$ 1 - \left( \frac{R}{R + h} \right)^2 = \frac{d}{R} $$.
Answer for screen readers
The relationship between height $h$ and depth $d$ for changes in gravitational acceleration is given by:
$$ 1 - \left( \frac{R}{R + h} \right)^2 = \frac{d}{R} $$
Steps to Solve
- Define the gravitational acceleration at height
The gravitational acceleration at a height $h$ above the Earth's surface can be given by the formula:
$$ g_h = g_0 \left( \frac{R}{R + h} \right)^2 $$
where $g_0$ is the acceleration due to gravity at the Earth's surface, and $R$ is the radius of the Earth.
- Define the gravitational acceleration at depth
The gravitational acceleration at a depth $d$ below the Earth's surface is modeled by:
$$ g_d = g_0 \left( 1 - \frac{d}{R} \right) $$
This relation assumes that the Earth's density is uniform.
- Determine the change in gravitational acceleration at height
To find the change in gravitational acceleration when moved to a height $h$, we calculate the difference:
$$ \Delta g_h = g_0 - g_h = g_0 - g_0 \left( \frac{R}{R + h} \right)^2 $$
- Determine the change in gravitational acceleration at depth
Similar to height, we can find the change in gravitational acceleration when moved to a depth $d$:
$$ \Delta g_d = g_0 - g_d = g_0 - g_0 \left( 1 - \frac{d}{R} \right) $$
- Express changes in terms of $R$
Both changes can be expressed in terms of the Earth's radius $R$:
Changing for height:
$$ \Delta g_h = g_0 \left[ 1 - \left( \frac{R}{R + h} \right)^2 \right] $$
Changing for depth:
$$ \Delta g_d = g_0 \left( \frac{d}{R} \right) $$
- Compare the two cases
To understand the relationship between the changes due to height and depth, set the two equations for changes equal to one another:
$$ g_0 \left[ 1 - \left( \frac{R}{R + h} \right)^2 \right] = g_0 \left( \frac{d}{R} \right) $$
This equation can help us derive a relation between $h$ and $d$.
The relationship between height $h$ and depth $d$ for changes in gravitational acceleration is given by:
$$ 1 - \left( \frac{R}{R + h} \right)^2 = \frac{d}{R} $$
More Information
This relationship shows that the decrease in gravitational acceleration due to an increase in height above the Earth's surface is related to the increase in depth below the surface. The formulas reflect how gravitational forces vary with distance from the Earth's center, illustrating fundamental principles in physics.
Tips
- Confusing the formulas for gravitational acceleration at height and depth: Ensure to remember that they have different forms.
- Not using the correct radius for calculating changes: Ensure to take Earth's radius into consideration for accurate results.
- Forgetting to equate changes properly when comparing height and depth.
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